### Abstract

Let C_{n} be the set of Dyck paths of length n. In this paper, by a new auto- morphism of ordered trees, we prove that the statistic 'number of exterior pairs', introduced by A. Denise and R. Simion, on the set C_{n} is equidistributed with the statistic 'number of up steps at height h with h ≡ 0 (mod 3)'. Moreover, for m ≥ 3, we prove that the two statistics 'number of up steps at height h with h ≡ 0 (mod m)' and 'number of up steps at height h with h ≡ m - 1 (mod m)' on the set C_{n} are 'almost equidistributed'. Both results are proved combinatorially.

Original language | English |
---|---|

Journal | Electronic Journal of Combinatorics |

Volume | 18 |

Issue number | 1 |

Publication status | Published - 2011 May 12 |

### Fingerprint

### Keywords

- Continued fraction
- Dyck path
- Exterior pair
- Ordered tree
- Planted tree

### ASJC Scopus subject areas

- Theoretical Computer Science
- Geometry and Topology
- Discrete Mathematics and Combinatorics
- Computational Theory and Mathematics
- Applied Mathematics

### Cite this

*Electronic Journal of Combinatorics*,

*18*(1).

**Exterior pairs and up step statistics on Dyck paths.** / Eu, Sen-Peng; Fu, Tung Shan.

Research output: Contribution to journal › Article

*Electronic Journal of Combinatorics*, vol. 18, no. 1.

}

TY - JOUR

T1 - Exterior pairs and up step statistics on Dyck paths

AU - Eu, Sen-Peng

AU - Fu, Tung Shan

PY - 2011/5/12

Y1 - 2011/5/12

N2 - Let Cn be the set of Dyck paths of length n. In this paper, by a new auto- morphism of ordered trees, we prove that the statistic 'number of exterior pairs', introduced by A. Denise and R. Simion, on the set Cn is equidistributed with the statistic 'number of up steps at height h with h ≡ 0 (mod 3)'. Moreover, for m ≥ 3, we prove that the two statistics 'number of up steps at height h with h ≡ 0 (mod m)' and 'number of up steps at height h with h ≡ m - 1 (mod m)' on the set Cn are 'almost equidistributed'. Both results are proved combinatorially.

AB - Let Cn be the set of Dyck paths of length n. In this paper, by a new auto- morphism of ordered trees, we prove that the statistic 'number of exterior pairs', introduced by A. Denise and R. Simion, on the set Cn is equidistributed with the statistic 'number of up steps at height h with h ≡ 0 (mod 3)'. Moreover, for m ≥ 3, we prove that the two statistics 'number of up steps at height h with h ≡ 0 (mod m)' and 'number of up steps at height h with h ≡ m - 1 (mod m)' on the set Cn are 'almost equidistributed'. Both results are proved combinatorially.

KW - Continued fraction

KW - Dyck path

KW - Exterior pair

KW - Ordered tree

KW - Planted tree

UR - http://www.scopus.com/inward/record.url?scp=79955734150&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=79955734150&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:79955734150

VL - 18

JO - Electronic Journal of Combinatorics

JF - Electronic Journal of Combinatorics

SN - 1077-8926

IS - 1

ER -